Exact and useful optimization methods for microeconomics

Abstract

This paper points out that the treatment of utility maximization in current textbooks on microeconomic theory is deficient in at least three respects: breadth of coverage, completeness-cum-coherence of solution methods and mathematical correctness. Improvements are suggested in the form of a Kuhn-Tucker type theorem that has been customized for microeconomics. To ensure uniqueness of the optimal solution stringent quasiconcavity, an apparently new adaptation of the notion of strict quasiconcavity, is introduced. It improves upon an earlier notion formulated by Aliprantis, Brown and Burkinshaw. The role of the domain of differentiability of the utility function is emphasized. This is not only to repair a widespread error in the microeconomic literature but also to point out that this domain can be chosen sensibly in order to include the maximization of certain nondifferentiable utility functions, such as Leontiev utility functions. To underscore the usefulness of the optimality conditions obtained here, five quite different instances of utility maximization are completely solved by a single coherent method.